Dynamic testing of nonlinear vibrating structures using nonlinear normal modes

ll rights reserved. +3243669505. s), [email protected] (G. Kerschen), [email protected] (J.C. Golinval). Finite element model ̈ M x (t)+ Kx (t) + fnl {x (t)} = 0 0 2 4 6 8 10 −100 −50 0 50 100 Time (s) A cc . ( m /s 2 ) Numerical NNM computation Experimental NNM extraction NNM frequencies NNM modal curves Experimental response (time series) Fig. 1. Theoretical and experimental nonlinear modal analysis. M. Peeters et al. / Journal of Sound and Vibration 330 (2011) 486–509 487 As reported in [10], a large body of literature exists regarding dynamic testing and identification of nonlinear structures, but very little work addresses nonlinear phenomena during modal survey tests. Interesting contributions in this context are [11–13]. The force appropriation of nonlinear systems (FANS) method extends linear force appropriation to nonlinear structures [11]. A multi-exciter force pattern that includes higher harmonic terms is used to counteract nonlinear coupling terms, which prevent any response other than the linear normal mode of interest. The nonlinear resonant decay (NLRD) method applies a burst of a sine wave at the undamped natural frequency of a linear mode and enables small groups of modes coupled by nonlinear forces to be excited [12]. A nonlinear curve fit in modal space is then carried out using the restoring force surface (RFS) method. Another test strategy that identifies nonlinearities in modal space using the RFS method is discussed in [13]. Alternatively, a nonlinear modal identification approach based on the single nonlinear resonant mode concept [14,15] and on a first-order frequency-domain approximation is proposed and applied in [16–19]. The forced frequency responses are expressed as a combination of a resonant nonlinear mode response and of linear contributions from the remaining modes. By a curve-fitting procedure, the amplitude-dependent nonlinear modal parameters may be identified from experimental responses close to the resonance. In this paper, an attempt is made to extend experimental modal analysis to a practical nonlinear analog using the nonlinear normal mode (NNM) theory. NNMs offer a solid and rigorous mathematical tool for analyzing nonlinear oscillations, yet they have a clear conceptual relation to the classical linear normal modes (LNMs) [20,21]. Another appealing feature of NNMs is that they are capable of handling strong structural nonlinearity. Following the philosophy of force appropriation, the proposed method excites the NNMs of interest, one at a time. To this end, the phase lag quadrature criterion is generalized to nonlinear structures in order to locate single-NNM responses. Thanks to the invariance principle (i.e., if the motion is initiated on one specific NNM, the remaining NNMs remain quiescent for all time), the energy dependence of the NNM modal curves and their frequencies of oscillation can be extracted directly from experimental time data. When used in conjunction with the numerical computation of the NNMs introduced in [22], the approach described herein leads to an integrated methodology for modal analysis of nonlinear vibrating structures (see Fig. 1). This methodology can, for instance, be used for validating dynamic models, as is routinely carried out for aerospace structures (e.g., ground vibration testing of aircrafts [23,24]). The present paper is organized as follows. In the next section, the definition of an NNM is briefly recalled. The theoretical modal analysis consisting of the numerical NNM computation from a finite element model is also introduced. In Section 3, the proposed procedure for nonlinear EMA is then presented. The different concepts are illustrated using a nonlinear two-degree-of-freedom (2DOF) system. Finally in Section 4, the approach is demonstrated and assessed using numerical experiments carried out on a nonlinear beam. 2. Theoretical modal analysis using nonlinear normal modes A detailed description of NNMs and of their fundamental properties (e.g., frequency-energy dependence, bifurcations and stability) is given in [20,21]. For completeness, the definition of an NNM is briefly reviewed in this section. The free response of discrete conservative mechanical systems with n degrees of freedom (DOFs) is considered, assuming that continuous systems (e.g., beams, shells or plates) have been spatially discretized using the finite element M. Peeters et al. / Journal of Sound and Vibration 330 (2011) 486–509 488 method. The general equations of motion are M € xðtÞþKxðtÞþfnlfxðtÞg 1⁄4 0 (1) where M is the mass matrix; K is the stiffness matrix; x, _ x and € x are the displacement, velocity and acceleration vectors, respectively; fnl is the nonlinear restoring force vector, including stiffness terms only. In the present study, as in [21], an extension of Rosenberg’s definition [25–27] is considered. An NNM motion is therefore defined as a (non-necessarily synchronous) periodic motion of the undamped mechanical system (1). This NNM definition may appear restrictive in case of nonconservative systems. However, as shown in this paper and in [21], the damped dynamics can often be interpreted based on the topological structure of the NNMs of the underlying conservative system. As evidenced in [22], this extended definition is particularly attractive when targeting a numerical computation of the NNMs. The approach followed here for the theoretical modal analysis consists in the numerical computation of undamped NNMs of nonlinear structures discretized by finite elements and governed by (1). The numerical method for the NNM computation relies on two main techniques, namely a shooting procedure and a method for the continuation of periodic solutions (i.e., NNM motions). A detailed description of the numerical algorithm is given in [22]. The NNMs are then obtained accurately, even in strongly nonlinear regimes, and in a fairly automatic manner. One typical dynamical feature of nonlinear systems is the frequency-energy dependence of their oscillations. As a result, the modal curves and frequencies of NNMs depend on the total energy in the system. For illustration, a 2DOF system with a cubic stiffness is considered in this paper. The conservative system is depicted in Fig. 2, and its motion is governed by the equations € x1þð2x1 x2Þþ0:5x1 1⁄4 0 € x2þð2x2 x1Þ 1⁄4 0 (2) Fig. 3 shows the time series, the configuration space, the power spectral density (PSD) and two-dimensional projections of the phase space of three in-phase NNM motions of increasing energies. The NNM motion at low energy resembles that of the in-phase LNM of the underlying linear system. The modal curve (i.e., the NNM motion in the configuration space) is a straight line, there is one main harmonic component in the system response, and the motion in phase space is an ellipse. For the motion at moderate energy, the NNM is now a curve, and the presence of two harmonic components can be detected in the PSD. A clear departure from the LNM (harmonic) motion is observed. At high energy, this is even more enhanced. For instance, the motion in phase space is a strongly deformed ellipse. When moving from the lowto the high-energy NNM, the period of the motion decreases from 6.28 to 4.755 s. This is due to the hardening characteristic of the cubic spring. Due to frequency-energy dependence, the representation of NNMs in a frequency-energy plot (FEP) is particularly convenient. An NNM motion is represented by a point in the FEP, which is drawn at the fundamental frequency of the periodic motion and at the conserved total energy during the motion, which is the sum of the potential and kinetic energies. A branch, represented by a solid line, is a family of NNM motions possessing the same qualitative features. For illustration, the conservative 2DOF system (2) is considered. The underlying linear system possesses two (in-phase and out-of-phase) LNMs. The FEP, computed using the numerical algorithm, is shown in Fig. 4. NNM motions in the configuration space (i.e., the modal curves) are inset. The backbone of the plot is formed by two branches, which represent in-phase (S 11+) and out-of-phase (S 11 ) synchronous NNMs. The indices in the notations are used to mention that the two masses vibrate with the same dominant frequency. These fundamental NNMs are the direct nonlinear extension of the corresponding LNMs. The FEP clearly shows that the nonlinear modal parameters, namely the modal curves and the corresponding frequencies of oscillation, have a strong dependence on the total energy in the system. The frequency of both the in-phase and out-of-phase NNMs increases with the energy level, which reveals the hardening characteristic of the cubic stiffness nonlinearity in the system. Additional branches corresponding to internally resonant NNMs, as opposed to fundamental NNMs, bifurcate from the backbone at higher energy as evidenced in [21]. However, these modal interactions occurring through internal resonances are beyond the scope of the present study. 1 1 1 1 x1 x2 1 0.5 Fig. 2. Schematic representation of the 2DOF system example.